Transcript Chapter 9

CHAPTER 9
Tangents, Arcs, and
Chords
SECTION 9-1
Basic Terms
CIRCLE
is the set of points in a
plane at a given
distance from a given
point in that plane. The
given point is the
CENTER of the circle
and the given distance
is the RADIUS.
CHORD
is a segment whose
endpoints lie on a
circle.
SECANT
is a line that contains
a chord.
DIAMETER
is a chord that
contains the center
of a circle.
TANGENT
is a line in the plane of
a circle that
intersects the circle
in exactly one point,
called the point of
tangency.
SPHERE
is the set of all points
in space at a
distance r (radius)
from point O (center)
CONGRUENT CIRCLES
Are circles that have
congruent radii
CONGRUENT SPHERES
Are spheres that have
congruent radii
CONCENTRIC CIRCLES
Are circles that lie in
the same plane and
have the same center
CONCENTRIC SPHERES
Are spheres that have
the same center.
INSCRIBED in a CIRCLE
Occurs when each
vertex of a polygon
lies on the circle and
the circle is
CIRCUMSCRIBED
about the POLYGON
SECTION 9-2
Tangents
THEOREM 9 -1
If a line is tangent to a
circle, then the line is
perpendicular to the
radius drawn to the
point of tangency.
Corollary
Tangents to a circle
from a point are
congruent
THEOREM 9 -2
If a line in the plane of a
circle is perpendicular
to the radius at its
outer endpoint, then
the line is tangent to
the circle.
CIRCUMSCRIBED about the
CIRCLE
Occurs when each side
of a polygon is
tangent to a circle
and the circle is
INSCRIBED in the
POLYGON
COMMON TANGENT
a line that is tangent
to each of two
coplanar circles
COMMON Internal TANGENT
Intersects the segment
joining the centers of
the circles.
COMMON External TANGENT
Does not intersect the
segment joining the
centers of the circles.
TANGENT CIRCLES
are coplanar circles
that are tangent to
the same line at the
same point
SECTION 9-3
Arcs and Central
Angles
CENTRAL ANGLE
is an angle with its
vertex at the center
of the circle.
ARC
is an unbroken part of
a circle.
MINOR ARC
is the arc formed by
two points on a circle
*The measure of a minor
arc is the measure of
its central angle
MAJOR ARC
is the remaining arc
formed by the
remaining points on
the circle
* The measure is 360°
minus the minor arc
SEMICIRCLE
is an arc formed from
the endpoints of a
circle’s diameter
* The measure is 180°
ADJACENT ARCS
Arcs that have exactly
one point in common.
CONGRUENT ARCS
Arcs in the same circle
or in congruent
circles that have
equal measure.
POSTULATE 16
The measure of the arc
formed by two
adjacent arcs is the
sum of the measures
of these two arcs
THEOREM 9-3
In the same circle or in
congruent circles,
two minor arcs are
congruent if and only
if their central angles
are congruent.
SECTION 9-4
Arcs and Chords
THEOREM 9-4
In the same circle or in
congruent circles:
1. Congruent arcs have
congruent chords
2. Congruent chords have
congruent arcs
THEOREM 9-5
A diameter that is
perpendicular to a
chord bisects the
chord and its arc
THEOREM 9-6
1.
2.
In the same circle or in
congruent circles:
Chords equally distant
from the center (or
centers) are congruent
Congruent chords are
equally distant from the
center (or centers)
SECTION 9-5
Inscribed Angles
INSCRIBED ANGLE
Is an angle whose
vertex is on a circle
and whose sides
contain chords of the
circle.
INTERCEPTED ARC
Is the intersection of
the sides of an
inscribed angle and
the circle
THEOREM 9-7
The measure of an
inscribed angle is
equal to half the
measure of its
intercepted arc
COROLLARY 1
If two inscribed angles
intercept the same
arc, then the angles
are congruent
COROLLARY 2
An angle inscribed in a
semicircle is a right
angle.
COROLLARY 3
If a quadrilateral is
inscribed in a circle,
then its opposite
angles are
supplementary
THEOREM 9-8
The measure of an
angle formed by a
chord and a tangent
is equal to half the
measure of the
intercepted arc.
SECTION 9-6
Other Angles
THEOREM 9-9
the measure of an angle
formed by two chords
that intersect inside a
circle is equal to half
the sum of the
measures of the
intercepted arcs.
THEOREM 9-10
the measure of an angle
formed by two secants,
two tangents, or a secant
and a tangent drawn from
a point outside a circle is
equal to half the
difference of the
measures of the
intercepted arcs
SECTION 9-7
Circles and Lengths of
Segments
THEOREM 9-11
When two chords
intersect inside a
circle, the product of
the segments of one
chord equals the
product of the
segments of the other
chord
THEOREM 9-12
When two secant segments
are drawn to a circle from
an external point, the
product of one secant
segment and its external
segment equals the
product of the other
secant segment and its
external segment.
THEOREM 9-13
When a secant and a
tangent segment are
drawn to a circle from an
external point, the
product of the secant
segment and its external
segment is equal to the
square of the tangent
segment
END